问题 解答题

设x=-1是f(x)=(x2+ax+b)e2-x(x∈R)的一个极值点,

(1)求a与b的关系式(用a表示b)并求f(x)的单调区间

(2)是否存在实数m,使得对任意a∈(-2,-1)及λ1λ2∈[-2,1]总有|f(λ1)-f(λ2)|<[(m+2)a+1]e3恒成立,若存在求出m的范围.若不存在,说明理由.

答案

(1)f'(x)=-[x2+(a-2)x+b-a]e2-x

由f'(-1)=0得b=2a-3…(2分)∴f(x)=(x2+ax+2a-3)e2-x

f′(x)=-[x2+(a-2)x+a-3]e2-x=-(x+1)(x+a-3)e2-x
令f′(x)=0得x1=-1,x2=3-a

由于x=-1是f(x)的极值点,故x1≠x2,即a≠4

①当a<4时,x2>x1,故[-1,3-a]为f(x)的单调增区间;(-∞,-1]、[3-a,+∞)为f(x)

的单调减区间.…(4分)

②当a>4时,x2<x1,故[[3-a,-1]为f(x)的单调增区间;(-∞,3-a]、[-1,+∞)为f(x)的单调减区间…(6分)

(2)由-2<a<-1得4<3-a<5,从而知f(x)在[-2,-1]单调递减,在[-1,1]上单调递增,

f(x)的值域为[f(-1),max{f(-2),f(1)}]=[(a-2)e3,e4]…(8分)

假设存在实数m满足题设,依题意有:[(m+2)a+1]e3>e4-(a-2)e3恒成立,

即(m+3)a-e-1>0恒成立,…(12分)

令g(a)=(m+3)a-e-1,

则有

g(-2)≥0
g(-1)≥0
,解得
m≤-
1
2
(e+7)
m≤-4-e
,即m≤-4-e

故存在实数m∈(-∞,-4-e]满足题设.…(14分)

多项选择题
阅读理解

In my long years of teaching, I often ask my student to read the texts until they can recite them, and I tell them that recitation is an important part of their homework. However, some of my students are tired of doing so, thinking that it is both hard and useless. They argue that it takes a long time to recite a text from memory and text itself never appears in a test paper.

In my opinion, it naturally takes time to recite a text, but it is worth doing so. When you are able to recite a text, you are sure to be familiar with the words and expressions. When you need them in reading or writing or doing exercises, they will come to your mind quickly, and so you will give quick response to all kinds of language situation. How can we say that it is useless?

Besides, many students complain that they soon forget what they were once able to recite. This is true, but you don’t need to worry. While you are reading and forgetting the texts, your language level is being raised. Sure you don’t remember your Chinese texts that you once read in the primary school, but now you are able to read novels and newspaper. So, recitation of the texts will help you improve your language ability.

61. The writer often asks his students to recite the texts,_______.

A.But they don’t like to do so.

B.And they are eager to do so

C.But they wrote a letter to the Department of Education reporting the thing.

D.But they do more exercises instead of reciting the texts.

62. What do the students think about the teacher’s advice?

A.The English teacher tries his best to help them with English.

B.The English teacher is very kind to them.

C.The English teacher asks them to do something useless.

D.The English teacher should let them study on their own.

63. According to the text, which of the following is correct?

A.The teacher thinks the students are too naughty to learn English well.

B.The students think they spend a lot of time in reciting the texts and get nothing.

C.The students say that they often forget the words.

D.The students take no interest in English.

64. When the students are against his suggestion, the writer_______.

A.often criticizes them

B.patiently explains to them why they should recite the texts

C.wants to give up his idea

D.is afraid that he is criticized by the headmaster.

65. Why does the writer suggest that the students recite the text?

A.Reciting the texts can improve the ability of applying English.

B.Reciting the texts is the best way of learning English.

C.The students should remember all the texts that the teacher has taught them.

D.Reciting the texts can greatly improve the classroom teaching.